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Ratio of the integral of x g(x) to the integral of g(x) when g is symmetric about the midpoint

This type of question, asked 2 times: 27 Jan 2024, 31 Jan 2024. All 2 questions below, oldest first.

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  1. Question 1

    27 Jan 2024

    64
    If (a,b)(a,b) be the orthocentre of the triangle whose vertices are (1,2),(2,3)(1,2),(2,3) and (3,1)(3,1), and I1=∫abxsin(4x−x2)dx, I2=∫absin⁡(4x−x2)dxI_{1}=\int_{a}^{b} xsin\left( 4x -x^{2} \right)dx,\ I_{2}=\int_{a}^{b} \sin\left( 4x -x^{2} \right)dx , then 36I1I236\frac{I_{1}}{I_{2}} is equal to :

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  2. Question 2

    31 Jan 2024

    90
    Let f:R→Rf:R \rightarrow R be a function defined by f(x)=4x4x+2f(x) =\frac{4^{x}}{4^{x}+ 2} and M=∫f(a)f(1−a)xsin⁡4(x(1−x))dxM =\int_{f(a)}^{f(1 - a)} x\sin^{4}(x(1 - x))dx, N=∫f(a)f(1−a)sin⁡4(x(1−x))dx;a≠12N =\int_{f(a)}^{f(1 - a)} \sin^{4}(x(1 - x))dx;a \neq\frac{1}{2}. If αM=βN,α,β∈N\alpha M = \beta N,\alpha,\beta \in N, then the least value of α2+β2\alpha^{2}+\beta^{2} is equal to